CONTROL SYSTEMS, ROBOTICS, AND AUTOMATION Vol. XI Model-Based Predictive Control - Edoardo Mosca

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1 COTROL SYSTEMS, ROBOTICS, AD AUTOMATIO Vol. XI Model-Based Predictive Control - Edoardo Mosca MODEL-BASED PREDICTIVE COTROL Edoardo Mosca University of Florence, Italy Keywords: optimal control, linear quadratic regulation, input saturations, state constraints, model uncertainties. Contents 1. Introduction 2. The Constrained Open-Loop Optimal Control (COLOC) Problem 3. Zero Terminal-State MBPC 4. Set-Membership Terminal Constraint 5. Time-Varying Ellipsoidal Terminal Constraint 6. Models, Disturbances and Robustness 7. Predictive Command Governors 8. Conclusive Remarks Glossary Bibliography Biographical Sketch Summary MBPC is a feedback-control methodology suitable to enforce efficiently hard constraints on the variables of the controlled system. It is shown that the method hinges upon a constrained open-loop optimal control problem along with the adoption of the so-called receding-horizon control strategy. In the important case of time-invariant linear saturated ACBI systems, MBPC algorithms can be devised with the property of ensuring global feasibility/stability. Considerations on how to deal with disturbances and model uncertainties are also given. A presentation of a simplified form of MBPC, viz. the PCG, is finally discussed. 1. Introduction UESCO EOLSS Model-Based Predictive Control (MBPC) is conceptually a natural method for generating feedback control actions for linear and nonlinear plants subject to pointwise-in-time input and/or state-related constraints. A human being, for instance, while driving a vehicle, generates steering-wheel commands, by forecasting or predicting over a finite time-horizon, the (possible) vehicle state-evolutions, on the basis of vehicle current state and dynamics, and a virtual or potential steering-wheel command sequence. Then, one, among such sequences, is sorted out, which fulfills safety constraints and meets performance requirements. Only a short initial portion of such a sequence is applied by the driver to the steering wheel, while its remaining part is discarded. After such an initial portion is applied, the driver repeats the whole operation by restarting predictions over a or receded time-horizon from the updated vehicle state as determined by the applied command. MBPC complies with the same logical scheme: the control

2 COTROL SYSTEMS, ROBOTICS, AD AUTOMATIO Vol. XI Model-Based Predictive Control - Edoardo Mosca sequence is computed by solving online, over a finite control horizon, an open-loop optimal control problem, given the plant dynamical model and current state. Though this computation hinges upon an open-loop control problem, MBPC yields a feedback-control action. Indeed, similarly to the driver behavior, in a discrete-time setting, only the first control of the open-loop control sequence is applied to the plant, and, according to the receding horizon control philosophy, the whole optimization cycle is repeated at the subsequent time-instant, based on the new plant-state. Because it involves a control horizon made up by only a finite number of time-steps, MBPC can be often calculated online, by existing optimization routines, so as to minimize a performance index, in the presence of hard constraints on the time evolutions of input and/or state. MBPC ability of handling constraints is of paramount importance whenever constraints are part of the control design specifications. In fact, constraints are typically present in applications, as they stem from actuators saturations and/or physical, safety or economical requirements. Despite the importance of constraints, there is a shortage of control methods for handling them effectively. The main reason for the interest of control engineers in MBPC is therefore its ability to systematically and effectively handle hard constraints. An important observation in this connection is that, in contrast to MBPC, in feedback-control systems of more traditional type, e.g., LQG or H control, constraints are indirectly enforced, by imposing, whenever possible, a conservative behavior at a performance-degradation expense. Other instances where MBPC can be advantageously used comprise unconstrained plants for which offline computation of a control law is a difficult task as compared with online computations via receding-horizon control. MBPC appears to have been proposed independently by several people, more or less simultaneously. It is not an easy matter to trace back its origin by looking at dates of related publications as the pioneers were mostly control practitioners who implemented MBPC well before the first publications appeared in the late seventies/early eighties. The early motivations for proposing MBPC were essentially twofold. On one side, the emphasis was on a control methodology, which would be applied to problems for which standard industrial controllers, e.g., PID, were inadequate. UESCO EOLSS Such a control methodology had to be based on intuitive concepts and offer ease of tuning. Within this context, constraint handling and optimality were not the main goals. On the other side, the emphasis was on an optimal plant operation under constraints, and control signal computations by repeatedly solving in real-time linear programming problems. To what nowadays we call MBPC, early publications gave various names, e.g., Dynamic Matrix Control, Model Predictive Heuristic Control, Receding Horizon Feedback Control, Heuristically Enhanced Feedback Control. A few patents related to early MBPC techniques were released starting from Among the advanced control methodologies, MBPC is the one, which has made the most significant impact on industrial control engineering. So far, it has been applied mainly in the petrochemical industry, even if it is being increasingly introduced in other sectors of the process industry. The main reason for its success in these applications are:

3 COTROL SYSTEMS, ROBOTICS, AD AUTOMATIO Vol. XI Model-Based Predictive Control - Edoardo Mosca 1. The processes are typically slow so that there is enough time for the online required computations; 2. MBPC can take into account actuator limitations; 3. Relatively to conventional control, MBPC allows operation closer to constraints, which often yields more profitable production. The success of MBPC may even become more significant in the future as an increasing use of nonlinear dynamic models is taking place in the process industry. These models are obtained by mathematically describing the chemical and physical transformations occurring inside the process. Then, there is a clear potential for future synergy of nonlinear models with MBPC in that nonlinear models can provide more accurate predictions of process behavior in a nonlinear regime. Significant use of MBPC has also been made in adaptive control during the last two decades. Adaptive control of non-minimum phase plants required in fact the use of underlying control laws more sophisticated than minimum-variance control but simple enough to be synthesized in real-time on the grounds of continuously updated plant identified models. Remarkable examples of MBPC use in adaptive control are GPC and MUSMAR, both developed during the eighties. The presentation of MBPC given hereafter aims at enlightening the main features of the approach, related well-established feasibility/stability constructive arguments, and current open problems. Consideration will be also given to the command governor, a specific control architecture of practical interest, which, though introduced independently of MBPC, in its recent developments has taken advantage of using conceptual tools of predictive control. For more specialized topics, the reader is referred to the three article level contributions dealing with MBPC, viz.: MBPC for Linear Systems ; MBPC for onlinear Systems ; and Adaptive Predictive Control. The presentation is organized as follows: Section 0 sets up the general ingredients of the constrained open-loop optimal control problem underlying any MBPC scheme. Section 0 describes the earliest and simplest form of a stabilizing MBPC algorithm. Section Error! Reference source not found. introduces a convenient form of a set-membership (ellipsoidal) terminal state-constraint devised so as to improve in terms of feasibility the algorithm of Section Error! Reference source not found.. UESCO EOLSS Section Error! Reference source not found. extends the scheme of Section Error! Reference source not found. by considering a state-dependent ellipsoidal constraint, which allows one to get global feasibility/stability whenever such a property is achievable in principle. Section Error! Reference source not found. describes how to deal with constant disturbances and nonzero setpoints, as well as model uncertainties of polytopic type. Section Error! Reference source not found. describes predictive reference governors. In Section Error! Reference source not found., a brief assessment of the current status of MBPC concludes the contribution. 2. The Constrained Open-Loop Optimal Control (COLOC) Problem In MBPC, the system to be controlled (plant) is usually represented by an ordinary

4 COTROL SYSTEMS, ROBOTICS, AD AUTOMATIO Vol. XI Model-Based Predictive Control - Edoardo Mosca differential equation. However, as the control is normally piecewise constant, the plant is most of the times described in terms of a difference equation x( k+ 1) = ϕ( x( k), u( k)) (1) yk ( ) = η( xk ( )) (2) n m p where x(k) is the state at time k, u(k) the input, y(k) a state-related vector connected to performance requirements (see Eq. (6) below), ϕ is assumed to be continuous at the origin with (ϕ (0 X, 0 U ) = 0 X ) and η (0 X ) = 0 Y. The plant input and state sequences are required to satisfy the constraints uk ( ) x( k) U (3) X (4) m where, usually, U is a convex and compact subset of, and X is a convex and closed n subset of, both sets containing the origin in their interior. For the event (x, t) (viz., for state x at time t), the cost is defined by t+ 1 J( x, t, u ) = l( x( k), u( k)) + L( x( t+ )) (5) k= t where u := {u(t), u(t + 1),, u(t + 1)} and x(k) = x u (k; (x, t)), the latter notation denoting the state at time k resulting from state x at time t k and a control sequence u. The terminal time t + increases with time t and, consequently, is referred to as a receding horizon. Various choices for the instantaneous loss l and the terminal loss L are in principle possible. However, according to the usual MBPC choice, hereafter the loss functions will be taken to be quadratic 2 2 lxk ( ( ), uk ( )): = yk ( ) ψ + uk ( ) y ψu 2 Lxt ( ( + )): = xt ( + ) ψ UESCO EOLSS 2 where v ψ : = v' ψ v, the prime denotes transpose, ψ y = ψ y ' > 0, ψ u = ψ u ' > 0 and ψ = ψ 0. In general, a terminal-state constraint ' (6) x( t+ ) X (7) is also imposed. At the event (x, t), the COLOC problem P (x, t) is to find, provided it exists, the optimal (virtual) control sequence u ( x, t) : = { u ( t; ( x, t)), u ( t+ 1; ( x, t)),..., u ( t+ 1; ( x, t))} (8)

5 COTROL SYSTEMS, ROBOTICS, AD AUTOMATIO Vol. XI Model-Based Predictive Control - Edoardo Mosca which minimizes J(x, t, u) subject to the control, state and terminal-state constraints, and yields the value function Vxt (, ): = Jxt (,, u ( xt, )) (9) According to the receding-horizon mode of operation, only the first control u (t; (x, t)) is applied to the plant input at time t. In such a way, a feedback-control action is obtained ut ( ) = cxt (, ): = u ( t;( xt, )) (10) Since ϕ (, ), η ( ), l(, ) and L( ) are time-invariant, problems P(x, t) are time-invariant in that V(x, t) = V (x, 0) and c(x, t) = c(x, 0). Consequently, it suffices at each event (x, t) to solve P (x) := P (x, 0). Problem P (x) is therefore as follows: P ( x): V ( x) = min{ J ( x, u) u U ( x)} (11) u 1 J ( x, u ) : = l( x( k), u( k)) + L( x( )) (12) k = 0 where u = {u(0), u(1),, u( 1)}, x(k) = x u (k; (x, 0)) and U (x) is the set of feasible control sequences, viz. sequences satisfying the control, state and terminal-state constraints. Because is finite, the minimum exists provided that ϕ (, ) and h( ) are continuous, U compact, X and X closed, and U (x) non-empty. At the event (x, t), P (x) is solved yielding the optimizing (virtual) control sequence u ( x) = { u (0; x), u (1; x),..., u ( 1; x)} (13) the optimal (virtual) state trajectory x ( x) = [ x (0; x) = x, x (1; x),..., x ( ; x)} (14) and the value function UESCO EOLSS V ( x) = J ( x, u ( x)) (15) The first control in the optimizing sequence u ( x) is applied to the plant input at the time t, and the MBPC action results c ( ) (0; ) x = u x (16) It is to be underlined that MBPC computes numerically online at the event (x, t) the optimal control action Eq. (16) rather than computing offline the optimal control law c ( ). It would be more convenient to explicitly compute offline, once for all, c ( ) via dynamic programming. As this is usually very hard of even impossible, MBPC computes at the

6 COTROL SYSTEMS, ROBOTICS, AD AUTOMATIO Vol. XI Model-Based Predictive Control - Edoardo Mosca event (x, t) the optimal control action c (x) rather than pre-computing the optimal control law c ( ). 3. Zero Terminal-State MBPC This can be regarded as the earliest and, conceptually, the simplest form of MBPC, which guarantees stability to the controlled system, whenever feasibility is satisfied. Here, X = {0 X } and, hence, the terminal-state constraint is the equality constraint x ( ) = 0 X (17) Given the plant state x, here U (x), the set of feasible control sequences, is the set of sequences, which drive the plant initial state x to the zero-state in steps with no constraint violation. Assume that U (x) is non-empty, and satisfaction of all other conditions for the existence of the optimizing control sequence u ( x) in Eq.(13). Then, asymptotic stability of the plant fed by the MBPC action can be easily proved by the following direct argument. Let u ( x) be the control sequence obtained from u ( x) by deleting its first control and inserting the zero-control 0 U in its final position u( x) : = { u (1; x), u (2; x),..., u ( 1; x), 0 } U This is a control sequence again of length which is feasible for x (1; x) = ϕ( x, u (0; x)). In fact, x (1; x) is driven by u( x) to 0 X in 1 steps, and held at 0 X at the -th step because (18) ϕ (0, 0 ) = 0 (19) X U X Specifically, the state trajectory over steps resulting from the initial state x (1; x) and the control sequence u( x) is x( x) : = { x (1; x),..., x ( 1; x), 0, 0 } Moreover, UESCO EOLSS J( x (1; x), u( x)) = J( x, u ( x)) l( x, u (0; x)) = V ( x) y(0) u(0) X X 2 2 ψ y ψ x (20) (21) where y(0) = η (x) is the initial plant output and u(0) = c (x) = u (0; x) is the effective input supplied at time 0 by MBPC to the plant. Then, if x(1) = ϕ (x, u(0)) denotes the effective plant state at time 1 in the MBPC-controlled system, according to Eq. (11) and Eq. (21) we get

7 COTROL SYSTEMS, ROBOTICS, AD AUTOMATIO Vol. XI Model-Based Predictive Control - Edoardo Mosca V ( x(1)) V ( x(0)) y(0) u(0) 2 2 ψ y ψ u Going to the generic time k, k = 0, 1,, for the closed-loop system we have 2 2 V ( x( k)) V ( x( k+ 1)) y( k) + u( k) (22) ψ y where u(k), x(k) and y(k) denote, respectively, the effective plant input, state and output at time k in the MBPC-controlled system. Eq. (22) shows that { V( x( k))} k= 0 is a monotonically non-increasing sequence. Hence, being V (x(k)) nonnegative, as k it converges to V (x( )), 0 V (x( )) V (x(0)). Consequently, summing both sides of Eq. (22) from k = 0 to k =, we get ψ u > V( x(0)) V( x( )) y( k) + u( k) This, in turn, implies as ψ y > 0 and ψ u > ψy ψu (23) k = 0 lim yk ( ) = 0 and lim uk ( ) = 0 (24) k Y k U Then, under a detectability condition on Eq. (1) and Eq. (2), for x U (x) 0/ one can conclude asymptotic stability of the closed-loop system: stability that can be seen to be of exponential type if the plant is linear, viz. x( k+ 1) =Φ x( k) + Gu( k) yk ( ) = Hxk ( ) The foregoing stability proof hinges upon the following two crucial points: (25) U ( ) 0 x / (26) u ( x) U ( x) u( x) U ( x (1; x)) (27) UESCO EOLSS ow, feasibility condition Eq. (26) can be lost in zero terminal-state MBPC, because of the need of driving the plant state to 0 X in a finite time. If the initial state is far from 0 X, this brings about the use of large plants inputs which can violate possible input saturation constraints. Various approaches can be adopted so as to enlarge the set of admissible states (states such that Eq. (26) is fulfilled). E.g., given a plant state x 0 X and an integer 1 such that U ( ) 0 1 x = /, one can always find, if the plant is controllable and only input saturation constraints are present, a possibly large but finite integer 2, 2 > 1, so as to make U ( ) 2 x non-empty. The disadvantage with this approach is that 2 can be too large, and, hence, the associated COLOC problem P ( ) 2 x too complex for an online solution for the available computing power and sampling time. Hereafter, forms of MBPC will be presented which are particularly tailored for both

8 COTROL SYSTEMS, ROBOTICS, AD AUTOMATIO Vol. XI Model-Based Predictive Control - Edoardo Mosca solving the feasibility problem and yielding a highly performing closed-loop system. In so doing, for the sake of simplicity, we shall address linear plants of the form Eq. (23) in the presence of only input-saturation constraints. The reason for the latter choice is that, while hard bounds on the manipulated variables are typically dictated by physical constraints (e.g., limited power of actuators): state-related constraints are frequently of the soft-type and, hence, can be properly addressed by penalizing additional state-related variables in the cost. The reason for concentrating hereafter on linear plants is to simplify the exposition and call the reader s attention more directly on features, e.g. set-membership terminal constraints, which are also equally important in MBPC of nonlinear systems Bibliography TO ACCESS ALL THE 23 PAGES OF THIS CHAPTER, Click here Allgöwer F. and Zheng A., eds. (2000). onlinear Model Predictive Control. Progress in Systems and Control Theory Series, 26. Basel, Switzerland: Birkhäuser Verlag. [A collection of papers originally presented at a workshop held in Ascona in 1998.] Alvarez-Ramírez J. and Suarez R. (1996). Global Stabilization of Discrete-Time Linear Systems with Bounded Inputs. International Journal of Adaptive Control and Signal Processing, 10, [This paper introduces and analyses the scheme of Section Error! Reference source not found. restricted to the case with no free moves ( = 0).] Angeli A., Casavola A. and Mosca E. (2000). Predictive PI-Control Under Positional and Incremental Input Saturations. Automatica,. 36, [This paper deals with MBPC control of linear systems under saturation constraints of both inputs and input increments.] Angeli D. and Mosca E. (1999). Command Governors for Constrained onlinear Systems. IEEE Transactions on Automatic Control. 44, [This paper extends the predictive command governor approach to nonlinear systems.] UESCO EOLSS Bemporad A., Casavola A. and Mosca E. (1997). onlinear Control of Constrained Linear Systems Via Predictive Reference Management. IEEE Transactions on Automatic Control. 42, [This paper describes the predictive approach to the command governor problem of deterministic linear systems with constraints.] Bemporad A., Chisci L. and Mosca E. (1994). On The Stabilizing Property of SIORCH. Automatica, 30, [This paper gives a proof of the stabilizing property of a variant of GPC.] Bemporad A. and Morari M. (1999). Control of Systems Integrating Logic, Dynamics, and Constraints, Automatica, 35, [This paper considers MBPC in cases where logic rules are part of the plant to be controlled under constraints.] Bemporad A., Morari M., Dua V. and Pistikopoulos E.. (2002). The Explicit Linear Quadratic Regulator for Constrained Systems. Automatica, 38, [This paper presents a technique for computing the explicit state-feedback solution to LQ control problems subject to state and input constraints.] Camacho E. and Bordons C. (1999). Model Predictive Control, 280 pp.. Berlin: Springer. [A monograph on MBPC theory and its applications.]

9 COTROL SYSTEMS, ROBOTICS, AD AUTOMATIO Vol. XI Model-Based Predictive Control - Edoardo Mosca Casavola A., Mosca E. and Angeli D. (2000). Robust Command Governors for Constrained Uncertain Linear Systems. IEEE Transactions on Automatic Control, 45, [This paper extends the predictive command governors approach to constrained linear systems with polytopic uncertainty models.] Casavola A., Giannelli M. and Mosca E. (1999). Globally Stabilizing Predictive Regulation of Input-Saturated Linear Systems. IEEE Transactions on Automatic Control, 44, [This paper introduces and analyses the globally feasible stabilizing MBPC scheme of Section Error! Reference source not found. with the state-dependent ellipsoidal terminal constraint.] Casavola A., Giannelli M. and Mosca E. (2000). Minimax Predictive Control Strategies for Input Saturated Polytopic Uncertain Systems. Automatica, 36, [This paper presents and analyses the min-max MBPC strategy of Section Error! Reference source not found. for input-saturated polytopic uncertainty models.] Chen C. C. and Shaw L. (1982). On receding Horizon Feedback Control. Automatica, 18, [This paper considers a nonlinear state-dependent receding-horizon regulation for time invariant linear plants so as to speed up the response to large regulation errors.] Chen H. and Allgöwer F. (1998). A Quasi Infinite-Horizon onlinear Model Predictive Control Scheme with Guaranteed Stability. Automatica, 34, [A paper which deals with nonlinear MBPC and its stability properties.] Chisci L., Lombardi A. and Mosca E. (1996). Dual Receding-Horizon Control of Constrained Discrete-Time Systems. European Journal of Control, 2, [This paper proposes an MBPC technique of the so-called dual type.] Clarke D. W., ed., (1994). Advances in Model Predictive Control. Oxford, U.K.: Oxford Science Publications. [A collection of papers presented in a workshop on MBPC held in Oxford in 1993.] Clarke D. W., Mohtadi C. and Tuffs P. S. (1987). Generalized Predictive Control. Part 1: The Basic Algorithms. Automatica, 23, [An introductory paper to the theory of GPC.] Clarke D. W. and Scattolini R. (1991). Constrained Receding Horizon Predictive Control, Proc. of the IEEE, Part D, Control Theory and Applications, 138, [This paper shows how to modify GPC so as to ensure stability for linear unconstrained systems.] Cutler C. R. and Ramaker B. L. (1980). Dynamic Matrix Control: A Computer Control Algorithm, Proc. Joint Automatic Control Conf., San Francisco, CA. [One of the earliest papers on MBPC with emphasis on optimal plant operation under constraints.] De icolao G., Magni L. and Scattolini R. (1996). On The Robustness of Receding Horizon Control with Terminal Constraints, IEEE Transactions on Automatic Control, 41, [A paper on stability robustness of MBPC algorithms ensuring nominal stability by terminal state constraints.] De icolao G., Magni L. and Scattolini R. (1996). Robust Predictive Control of Systems with Uncertain Impulse Response, Automatica, 32, [A paper on stability robustness of MBPC algorithms based on impulse response models affected by uncertainty.] UESCO EOLSS Gambier A. and Unbehauen H. (1999). Multivariable Generalized State-Space Receding Horizon Control in A Real-Time Enviroment, Automatica, 35, [A contribution to MBPC of multivariable plants described by state equations.] Greco C., Menga G., Mosca E. and Zappa G. (1984). Performance Improvement of Self-Tuning Controllers by Multistep Horizons: the MUSMAR Approach, Automatica, 20, [This paper introduces MUSMAR, a self-tuning controller embodying an MBPC algorithm based on a special identified prediction model of the plant under control.] Gilbert E. G. and Tin Tan K. (1991). Linear Systems with State and Control Constraints: The Theory and Applications of Maximal Output Admissible Sets, IEEE Transactions on Automatic Control, 36, [This paper introduces important basic concepts and results for designing command governors.] Kerthi S. S. and Gilbert E. G. (1988). Optimal Infinite-Horizon Feedback Laws for A General Class of Constrained Discrete-Time Systems: Stability and Moving-Horizon Approximations, Journal of Optimization Theory and Applications, 57, [This is a fundamental and seminal paper on methods

10 COTROL SYSTEMS, ROBOTICS, AD AUTOMATIO Vol. XI Model-Based Predictive Control - Edoardo Mosca of stability analysis for zero terminal-state MBPC of constrained discrete-time nonlinear systems.] Kothare M. V., Balakrishnan and Morari M. (1996). Robust constrained model predictive control using linear matrix inequalities, Automatica, 32, [This paper introduces the algorithm of Section Error! Reference source not found. restricted to the case with no free moves ( = 0).] Kwon W. and Pearson A. (1977). A Modified Quadratic Cost Problem and Feedback Stabilization of A Linear System, IEEE Transactions on Automatic Control, 223, [An early seminal paper on the zero terminal-state receding-horizon control of unconstrained linear systems.] Lee J. H. and Yu Z. (1997). Worst-Case Formulations of Model Predictive Control for Systems with Bounded Parameters, Automatica, 33, [Min-max formulations of MBPC for state-space plants with bounded parameters are considered in the paper.] Maciejowski J.M. (2002). Predictive Control: with Constraints, 331 pp. Harlow, Essex, England: Prentice Hall. [A book which brings together many aspects of MBPC from underlying concepts to likely future directions, both of research and applications.] Martín Sánchez J.M. (1976). Adaptive Predictive Control Systems. U.S. Patent o.4, 196, 576. Priority date: August 4, [The first patent related to the use of MBPC.] Mayne D. Q. and Michalska H. (1990). Receding Horizon Control of on-linear Systems, IEEE Transactions on Automatic Control, 35, [This paper deals with various issues arising in zero terminal-state MBPC of constrained continuous-time nonlinear systems.] Mayne D. Q., Rawlings J. B., Rao C. V. and Scokaert P. O. M. (2000). Constrained Model Predictive Control: Stability and Optimidity Automatica, 36, [A detailed and well-documented survey paper on MBPC.] Michalska H. and Mayne D. Q. (1993). Robust Receding Horizon Control of Constrained onlinear Systems, IEEE Transactions on Automatic Control, 38, [A contribution to continuous-time MBPC with constraints.] Michalska H. and Mayne D. Q. (1995). Moving Horizon Observers and Observer-Based Control, IEEE Transactions on Automatic Control, 40 (6), [This paper considers the use of receding horizon in state observers and observer-based control.] Mosca E. (1995). Optimal, Predictive, and Adaptive Control, 477 pp.. Englewood Cliffs, J: Prentice Hall. [This book deals with several problems of MBPC of deterministic or uncertain stochastic systems, particularly it describes and analyses the use of banks of parallel identifiers of multi-step ahead predictive models in adaptive (self-tuning) predictive control.] Mosca E., Lemos J. L. and Zhang J. (1990). Stabilizing I/O Receding Horizon Control, Proc. 29 th IEEE Conf. on Decision and Control, Honolulu, [An early version of Mosca E. and Zhang J. (1992).] Mosca E., Zappa G. and Lemos J.M. (1989). Robustness of Multipredictor Adaptive Regulators: MUSMAR, Automatica, 25, [This paper presents an analysis of the robustness properties of MUSMAR, an adaptive predictive controller.] UESCO EOLSS Mosca E. and Zhang J. (1992). Stable Redesign Of Predictive Control, Automatica, 28, [This paper deals with the problem of redesigning MBPC algorithms for I/O models so as to ensure stability to the controlled system.] Propoi A. I. (1963). Use of Linear Programming Methods for Synthesizing Sample-Data Automatic Systems, Automation and Remote Control, 24 (7), [One of the earliest proposal of a form of MBPC using linear programming for linear systems with hard constraints.] Rawlings J. B. (2000). Tutorial Overview of Model Predictive Control, IEEE Control Systems Magazine, 20, [An easy access to MBPC principles and its use in applications.] Rawlings J. B. and Muske K. R. (1993). The Stability of Constrained Receding-Horizon Control, IEEE Transactions on Automatic Control, 38 (10), [This paper reports one of the first attempts to overcome the feasibility problem in MBPC of linear systems with constraints.] Richalet J., Rault A., Testud J. L. and Papon J. (1978). Model Predictive Heuristic Control: Applications to Industrial Processes, Automatica, 14, [This is one of the earliest papers to advocate MBPC for

11 COTROL SYSTEMS, ROBOTICS, AD AUTOMATIO Vol. XI Model-Based Predictive Control - Edoardo Mosca process control.] Rossiter J. A., Kouvaritakis B. and Rice M. J. (1998). A umerically Robust State-Space Approach To Stable Predictive Control Strategies, Automatica, 34, [This paper presents a variant of the type of MBPC earlier discussed in (Mosca and Zhang, 1992).] Scokaert P. O. M. and Mayne D. Q. (1998). Min-Max Feedback Model Predictive Control for Constrained Linear Systems, IEEE Transactions on Automatic Control, 43, [This paper deals with the presence of persistent unpredictable disturbances in MBPC.] Scokaert P. O. M. and Rawlings J. B. (1998). Constrained Linear Quadratic Regulation, IEEE Transactions on Automatic Control, 43 (8), [This paper discusses the MBPC scheme of Section Error! Reference source not found. restricted to the case =.] Scokaert P. O. M., Rawlings J. B. and Meadows E. S. (1997). Discrete-Time Stability with Perturbations: Applications to Model Predictive Control, Automatica, 33(3), [A contribution on how to deal with persistent disturbances in MBPC.] Scokaert P. O. M., Mayne D. Q. and Rawlings J. B. (1999). Suboptimal Model Predictive Control (Feasibility Implies Stability). IEEE Transaction on Automatic Control, 44(3), [A contribution on the intimate relationship between feasibility and stability in MBPC with constraints.] Soeterboek R. (1992). Predictive Control: A Unified Approach, Englewood Cliffs, J: Prentice Hall. [An early monograph on MBPC.] Sznaier M. and Damborg M. J. (1990). Heuristically Enhanced Feedback Control of Constrained Discrete-Time Systems, Automatica, 26 (3), [A contribution along the lines of MBPC to the problem of controlling plants subject to constraints.] Thomas Y. A. (1975). Linear Quadratic Optimal Estimation and Control with Receding Horizon, Electronics Letters, 11, [This is one of the earliest papers to propose MBPC for unconstrained continuous-time systems.] Zheng A. and Morari M. (1995). Stability of Model Predictive Control with Mixed Constrains, IEEE Transactions on Automatic Control, 40 (10), [This paper discusses how to soften hard state constraints by adding extra terms in the quadratic performance index.] Biographical Sketch Edoardo Mosca obtained his Dr. Eng. degree in Electronics Engineering from the University of Rome La Sapienza. He then spent four years, from 1964 to 1968, in industry where he worked on the research and development of advanced radar systems. Thereafter, he held academic positions at the University of Michigan, Ann Arbor, Michigan, and McMaster University, Ontario, Canada. Since 1972, he has been with the Engineering Faculty, University of Florence, Italy: from 1972 to 1975 as an Associate Professor, and since 1975 as a full Professor of Control Engineering. He is the author of more than 140 research papers spanning several diversified fields such as radar signal synthesis and processing, radio communications, system identification, adaptive, predictive, and switching supervisory control. He is the author of a book, Optimal, Predictive, and Adaptive Control, Prentice Hall, He is also the editor of the following journals: European Journal of Control, International Journal of Adaptive Control and Signal Processing, Wiley, and IEE Proceedings-Control Theory and Applications. He is a Council member of EUCA (European Union Control Association); the Italian MO representative in IFAC (International Federation of Automatic Control); a Council member of IFAC; and a Fellow of the IEEE (Institute of Electrical and Electronics Engineers). UESCO EOLSS